Let $\overline{\mathrm{a}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and…

Let $\overline{\mathrm{a}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$. Then the vector $\overline{\mathrm{b}}$ satisfying $\overline{\mathrm{a}} \times \overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=3$, is
  1. $-\hat{i}+\hat{j}-2 \hat{k}$
  2. $2 \hat{i}-\hat{j}+2 \hat{k}$
  3. $\hat{i}-\hat{j}-2 \hat{k}$
  4. $\hat{i}+\hat{j}-2 \hat{k}$

Solution

$\begin{aligned} & \text { Given, } \overline{\mathrm{a}} \times \overline{\mathrm{b}}+\overline{\mathrm{c}}=0 \\ & \Rightarrow \overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})+\overline{\mathrm{a}} \times \overline{\mathrm{c}}=0 \\ & \Rightarrow(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{a}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{a}}) \overline{\mathrm{b}}+\overline{\mathrm{a}} \times \overline{\mathrm{c}}=0 \\ & \Rightarrow 3 \overline{\mathrm{a}}-2 \overline{\mathrm{~b}}+\overline{\mathrm{a}} \times \overline{\mathrm{c}}=0 \Rightarrow 2 \overline{\mathrm{~b}}=3 \overline{\mathrm{a}}+\overline{\mathrm{a}} \times \overline{\mathrm{c}} \\ & \Rightarrow 2 \overline{\mathrm{~b}}=3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}=-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}} \\ & \Rightarrow \overline{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\end{aligned}$

Asked in: MHT CET 2024 (03 May Shift 2)

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